Rutherford and the discovery of the nucleus

So, if the positive charge within an atom is spread out
evenly throughout its entire volume, as it Thomson's
plum-pudding model,
one can calculate the typical angle by which
an alpha particle might be deflected in an interaction
with a single atom:

theta = 0.01 degree

As an alpha particle moves through a thin section of foil,
it encounters many atoms. Each atom gives it a small
deflection in some random direction.
After it comes out of the foil, the alpha particle
has some overall deflection delta
which is the sum of all the small deflections.

In order to bend by a large angle,
say, 90 degrees,
all the small deflections would have to
be in the same direction:
all "to the left", for example.
That's not very likely.
If there's a 50% chance that each deflection
goes "to the left" versus "to the right",
what are the chances that consecutive deflections
add up to 90 degrees?

Q: How many consecutive deflections in a row
would it take to add up to delta=90 degrees overall?
Q: What are the odds that this happens to one
particular alpha particle?

In the early twentieth century,
English physicist Ernest Rutherford
ran a lab in which experiments of all kinds
were performed.
One of his assistants, Hans Geiger,
investigated scattering of alpha particles
by thin films of different metals.
In this paper from 1910,

Geiger reports on his work.
He set up an evacuated glass tube with a radioactive
material (radium) coating the walls of the tube at one
end, and a flourescent screen at the other end:

The conical section of glass, A,
was coated with radium.
After about fifteen minutes,
the most active radioactive materials disappear,
leaving only a single isotope
which emitted alpha particles.
These flew down the length of the tube,
through a small diaphragm D,
then struck a foil at E.
At the very end of the tube was a flourescent
screen S,
which would light up with a tiny flash
whenever an alpha particle struck it.
Geiger would very carefully count the number of
flashes at various points on the screen,
and derive the angle through which the alpha
particles had been deflected by the screen.

Now, Geiger used the Thomson model to interpret
his results.
He found a typical scattering angle which
was indeed small:

But there was one little surprise:
he occasionally saw a flash of light on a screen placed
on the OTHER side of the foil,
meaning that an alpha particle
had bounced BACKWARDS.

He determined that about 1 in 8000 alpha particles
bounced backwards.
But if the typical scattering angle is
just 1/200 of a degree, then it should take
a number of consecutive "left-hand" turns N given by

90 degrees
N = ------------- = 18,000
1/200 degree

If there's a 50% chance of scattering to the left or the right
each time the alpha encounters an atom, the probability of
18,000 consecutive left-hand turns is

-18,000 -6,000
prob P = 2 = 10

which is a lot smaller than 1 in 8000. A LOT smaller.
A REALLY BIG LOT smaller!

In the words of Rutherford,

It was quite the most incredible event that ever
happened to me in my life. It was as incredible
as if you fired a 15-inch shell at a piece of
tissue paper and it came back and hit you.

So what was going on? The Thomson model can't explain it.
In the old-fashioned tradition of being cautious
(which doesn't exist much anymore -- it doesn't get press),
Geiger writes:

But he (and Rutherford) did have an idea ...
one that they published three years later,
after an exhaustive series of similar experiments.

Rutherford's model: the positive nucleus

Rutherford realized that a bunch of weak scattering events
would never lead to the observed frequency of angles
greater than 90 degrees.
For one thing, the thin foils he and his colleagues
were using were only several hundred atoms thick;
there just wasn't time for an alpha particle
to encounter the thousands of atoms needed to
bend its path by large angles.

Perhaps each alpha underwent just one scattering event --
but one which could sometimes be very strong.

The experiments showed several relationships between
the number of alpha particles scattered
at some angle theta
and other quantities:
the type of material in the foil,
its thickness,
the kinetic energy of the incoming alpha particles,
etc.
Rutherford worked out a model which could
explain all the correlations.

From "The Scattering of alpha and beta Particles
by Matter and the Structure of the Atom,"
Philosophical Magazine, vol. 21, p. 669 (1911)

In this new model,
all the positive charge of the atom was concentrated
in a tiny point at the center;
around it was nearly empty space,
inhabited by the electrons alone.
As Rutherford put it, the atom wasn't
a plum pudding; it was more like

... like a few flies in a cathedral ...

The crucial point was the very small size of the
"nucleus":
since all the positive charge was concentrated in so small
a volume,
the projectile could OCCASIONALLY come very, very close to it;
so close that the electric force would be large enough to
push the projectile by a very large angle, maybe even backwards.

Let's use the impulse approximation again to get a feeling
for what's happening.

If the alpha particle approaches the nucleus to a minimum
distance R, then the impulse imparted to the alpha
by the electric force is roughly the force at closest
approach multipled by the duration of the passage:

Q: What happens to this impulse as the
distance R decreases?
Q: What is the rough size of an atom?
Q: What is the rough size of a nucleus?
Q: How much stronger is the impulse if the
positive charge is compressed within
the nucleus, rather than being spread
throughout the entire volume of an atom?

Using his model,
Rutherford derived two formulae which described the
behavior of alpha particles
as a function of the other variables
in the experiment.
First, he could determine the
impact parameter b
of an charged particle which was scattered
by an angle theta:

Given the Coulomb force constant ke,
the charge z (in electron units) on the projectile,
the charge Z (in electron units) on the atom,
and the kinetic energy K of the projectile,
Rutherford's model yields:

One can use this formula to derive equation 6.15 in
your textbook. That derived relationship concerning
the fraction of particles scattered by more or less
than a particular angle will come in handy on this week's
homework assignment.

If one measures the fraction of particles
which are scattered by various angles
as they pass through a foil,
one can work out the fraction of the cross-section
of an atom which is occupied by the nucleus.
In other words,
one can measure the size of the nucleus!

Rutherford's model also makes a series of predictions
between the fraction of particles scattered
at a particular angle
and a number of quantities involved in the experiment:

composition of foil

thickness of foil

kinetic energy of projectile particles

charge on the projectile particles

Rutherford was able to put all this information into a
single complicated formula:

Using this formula,
and the experimental results of Geiger and Marsden,
Rutherford was able to figure out the approximate
electric charge of the nucleus of a gold atom:

What's the currently accepted value for the electric
charge in a gold nucleus?

One of the somewhat surprising results of the experiments
was that the variations in angle didn't run
exactly as one might expect with elements of increasing
atomic weight.
Naively, one might expect that an element
with an atomic weight of N
might have N positive charges at its center.
However, as Geiger and Marsden write in the conclusion
to their paper
The Laws of Deflexion of alpha Particles
through Large Angles,
Philosophical Magazine, vol 25, p. 604 (1913):

Can you explain why the number of positive charges N
in a nucleus was only about HALF the atomic weight?

Summary of the scattering results, as of 1913

The positive charge and almost all the mass of an atom
is contained in a very compact nucleus,
the size of which is roughly 10^(-15) m.

One can estimate the electric charge within an atomic
nucleus roughly.

The positive charge of the nucleus (in multiples of the charge
of an electron)
is roughly equal to half the atomic weight
(in multiples of hydrogen's mass).

Quite an impressive amount of information, no?
As an analogy,
imagine trying to discern the internal structure of an automobile
from a distance of several hundred yards
by shooting a high-powered rifle at it.